Generating Tree Permutations
Constant-memory Iterative Generation of
Special Strings Representing Binary Trees
Sebastian Smyczy«ski
Faculty of Mathematics and Computer Science,
Nicolaus Copernicus University, Toru«, Poland
Sebastian Smyczy«ski
Generating Tree Permutations
Interesting combinatorical problem
Enumerate all possible shapes of binary trees with the given
number of nodes.
Sebastian Smyczy«ski
Generating Tree Permutations
Natural order
Natural order of binary trees
The natural order of binary trees follows the recursive denition:
We say that T ≺ T if:
1
2
|T1 | < |T2 |, or
|T1 | = |T2 | and left (T1 ) ≺ left (T2 ), lub
|T1 | = |T2 | and left (T1 ) = left (T2 ) and right (T1 ) ≺ right (T2 ),
Sebastian Smyczy«ski
Generating Tree Permutations
Natural order
Natural order of binary trees
The natural order of binary trees follows the recursive denition:
We say that T ≺ T if:
1
2
|T1 | < |T2 |, or
|T1 | = |T2 | and left (T1 ) ≺ left (T2 ), lub
|T1 | = |T2 | and left (T1 ) = left (T2 ) and right (T1 ) ≺ right (T2 ),
The size matters - the smaller trees precedes the larger ones.
Sebastian Smyczy«ski
Generating Tree Permutations
Natural order
Natural order of binary trees
The natural order of binary trees follows the recursive denition:
We say that T ≺ T if:
1
2
|T1 | < |T2 |, or
|T1 | = |T2 | and left (T1 ) ≺ left (T2 ), lub
|T1 | = |T2 | and left (T1 ) = left (T2 ) and right (T1 ) ≺ right (T2 ),
The size matters - the smaller trees precedes the larger ones.
It is dened in natural for binary trees recursive way.
Sebastian Smyczy«ski
Generating Tree Permutations
Natural order
Figure: All shapes of binary trees with 4 nodes listed in their natural order
Sebastian Smyczy«ski
Generating Tree Permutations
Tree permutations
Sebastian Smyczy«ski
Generating Tree Permutations
Tree permutations
We can represent the tree T with n nodes
uniquely as a sequence of the integer numbers
1, 2, . . . , n,
Sebastian Smyczy«ski
Generating Tree Permutations
Tree permutations
5
We can represent the tree T with n nodes
uniquely as a sequence of the integer numbers
1, 2, . . . , n,
rst labeling the nodes with their position's
number as they appear in the preorder
travelsal of the tree,
Sebastian Smyczy«ski
2
1
Generating Tree Permutations
7
3
6
4
Tree permutations
5
We can represent the tree T with n nodes
uniquely as a sequence of the integer numbers
1, 2, . . . , n,
rst labeling the nodes with their position's
number as they appear in the preorder
travelsal of the tree,
then listing those labels as they appear in the
preorder traversal of the tree.
Sebastian Smyczy«ski
2
1
7
3
4
5213476
Generating Tree Permutations
6
Tree permutations
5
We can represent the tree T with n nodes
uniquely as a sequence of the integer numbers
1, 2, . . . , n,
rst labeling the nodes with their position's
number as they appear in the preorder
travelsal of the tree,
then listing those labels as they appear in the
preorder traversal of the tree.
We shall call the resulting permutation
p = p , p , . . . , pn tree permutation of the T .
1
2
1
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4
5213476
2
Sebastian Smyczy«ski
7
Generating Tree Permutations
6
Tree permutations and the natural order
Figure: All shapes of binary trees with 4 nodes listed in their natural order
Sebastian Smyczy«ski
Generating Tree Permutations
Tree permutations and the natural order
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Figure: All shapes of binary trees with 4 nodes listed in their natural order
Sebastian Smyczy«ski
Generating Tree Permutations
Tree permutations and the natural order
1
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2
1
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3
1
4
4
1234
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1243
3
3
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1324
2
3124
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2
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1432
4
2134
3
2143
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4123
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1423
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3214
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4132
4
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1
3
4213
2
2
4312
1
4321
Figure: All shapes of binary trees with 4 nodes listed in their natural order
Sebastian Smyczy«ski
Generating Tree Permutations
Tree permutations and the natural order
1
1
2
2
3
1
4
4
1234
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1243
4
1324
4
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3
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2
3124
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4
1
2
3
3214
4123
1234
1243
1234
1
12434
1324
2
1423
3
1432
1423
2134
2143
31244
1
3214
41233
4132
2
4213
4132
4312
4321
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4
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2
1
2
3
2
1
4
4
1432
2134
3
2143
4
4
2
1
4
3
3
1
3
4213
2
2
4312
1
4321
Figure: All shapes of binary trees with 4 nodes listed in their natural order
Sebastian Smyczy«ski
Generating Tree Permutations
Recursive property of the tree permutations
∀p ∈ Tn
∃p 0 ∈ Tp −1 ∃p 00 ∈ Tn−p
p = p1 p 0 (p 00 ⊕ p1 )
1
1
p 00
4 3 1 2⊕
z
{ z
}|
{
p = 8 4 1 3 2 6 5 7 12 11 9 10
p0
}|
Sebastian Smyczy«ski
z
}|
{
Generating Tree Permutations
8
Climbing property of tree permutations
Lemma
A permutation p = p , p , . . . , pn of the integer numbers 1, 2, . . . , n is
a tree permutation i it is 231-avoiding - there are no such indices
i < j < k such that pk < pi < pj .
1
2
Sebastian Smyczy«ski
Generating Tree Permutations
Climbing property of tree permutations
Lemma
A permutation p = p , p , . . . , pn of the integer numbers 1, 2, . . . , n is
a tree permutation i it is 231-avoiding - there are no such indices
i < j < k such that pk < pi < pj .
1
2
pj
3
2
pi
1
Sebastian Smyczy«ski
pk
Generating Tree Permutations
Climbing property of tree permutations
Lemma
A permutation p = p , p , . . . , pn of the integer numbers 1, 2, . . . , n is
a tree permutation i it is 231-avoiding - there are no such indices
i < j < k such that pk < pi < pj .
1
2
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9
7
5
Sebastian Smyczy«ski
Generating Tree Permutations
Climbing property of tree permutations
Lemma
A permutation p = p , p , . . . , pn of the integer numbers 1, 2, . . . , n is
a tree permutation i it is 231-avoiding - there are no such indices
i < j < k such that pk < pi < pj .
1
2
15
14
12
10
9
7
5
Sebastian Smyczy«ski
Generating Tree Permutations
Working sux properties
Denition
Let p = p , p , . . . , pn be a tree
permutation. The sux of p which
makes p dierent from its successor in
the lexicographic order is called working
sux.
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2
Sebastian Smyczy«ski
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Generating Tree Permutations
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1
Working sux properties
Denition
Let p = p , p , . . . , pn be a tree
permutation. The sux of p which
makes p dierent from its successor in
the lexicographic order is called working
sux.
1
2
Lemma
Let p be a tree permutation and let i be
the index of the rst position of its
working sux. If tree permutation q is
the successor of p in the lexicographic
order, then qi = pi + 1.
Sebastian Smyczy«ski
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3
Generating Tree Permutations
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Working sux properties
Lemma
Let p be a tree permutation and
let i be the index of the rst
position of its working sux, then
there exist no such indices j, k
such that i < j < k and
pk = pj + 1.
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Lemma
Let p = p , p , . . . , pn be a tree
permutation and i be the starting
index of its working sux. For
any index i ≤ k < n, pk > pk + 1
implies that pk = pk + 1 + 1.
1
2
Sebastian Smyczy«ski
8
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4
3
2
1
Generating Tree Permutations
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
Generating Tree Permutations
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
Generating Tree Permutations
The algorithm
1 2 3 4
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
Generating Tree Permutations
The algorithm
1 2 3 4
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
Generating Tree Permutations
The algorithm
1 2 3 4
1 2 4 3
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
Generating Tree Permutations
The algorithm
1 2 3 4
1 2 4 3
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
Generating Tree Permutations
The algorithm
1 2 3 4
1 2 4 3
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
Generating Tree Permutations
The algorithm
1 2 3 4
1 2 4 3
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
Generating Tree Permutations
The algorithm
1 2 3 4
1 2 4 3
1 3 4 2
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
Generating Tree Permutations
The algorithm
1 2 3 4
1 2 4 3
1 3 4 2
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
Generating Tree Permutations
The algorithm
1 2 3 4
1 2 4 3
1 3 2 4
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
Generating Tree Permutations
The algorithm
1 2 3 4
1 2 4 3
1 3 2 4
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
Generating Tree Permutations
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
1
1
1
1
2
2
3
4
Generating Tree Permutations
3
4
2
2
4
3
4
3
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
1
1
1
1
2
2
3
4
Generating Tree Permutations
3
4
2
2
4
3
4
3
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
1
1
1
1
2
2
3
4
Generating Tree Permutations
3
4
2
2
4
3
4
3
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
1
1
1
1
2
2
3
4
Generating Tree Permutations
3
4
2
2
4
3
4
3
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
1
1
1
1
1
2
2
3
4
4
Generating Tree Permutations
3
4
2
2
3
4
3
4
3
2
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
1
1
1
1
1
2
2
3
4
4
Generating Tree Permutations
3
4
2
2
3
4
3
4
3
2
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
1
1
1
1
1
2
2
3
4
4
Generating Tree Permutations
3
4
2
2
3
4
3
4
3
2
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
1
1
1
1
1
2
2
3
4
4
Generating Tree Permutations
3
4
2
2
3
4
3
4
3
2
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
1
1
1
1
1
2
2
2
3
4
4
4
Generating Tree Permutations
3
4
2
2
3
3
4
3
4
3
2
1
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
1
1
1
1
1
2
2
2
3
4
4
4
Generating Tree Permutations
3
4
2
2
3
3
4
3
4
3
2
1
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
1
1
1
1
1
2
2
2
3
4
4
1
Generating Tree Permutations
3
4
2
2
3
3
4
3
4
3
2
4
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
1
1
1
1
1
2
2
3
3
4
4
4
4
4
2
2
3
4
4
1
1
1
2
1
1
2
3
3
Generating Tree Permutations
3
4
2
2
3
3
4
2
1
2
3
1
1
2
4
3
4
3
2
4
3
4
4
3
2
3
2
1
The algorithm
Generating tree permutations in
lexicographic order:
Start with permutation 1 2 . . . n.
Find the rst pair of indices i < j
starting from the end of the
permutation such that pj = pi + 1.
Swap the elements at found
positions.
Sort all but the rst elements of the
working sux in ascending order.
Sebastian Smyczy«ski
1
1
1
1
1
2
2
3
3
4
4
4
4
4
2
2
3
4
4
1
1
1
2
1
1
2
3
3
Generating Tree Permutations
3
4
2
2
3
3
4
2
1
2
3
1
1
2
4
3
4
3
2
4
3
4
4
3
2
3
2
1
Optimalization of the searching step
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Sebastian Smyczy«ski
Generating Tree Permutations
Optimalization of the searching step
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Sebastian Smyczy«ski
Generating Tree Permutations
Optimalization of the searching step
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9 6 3 2 1 5 4 7 8 11 10 22 19 17 13 12 16 15 14 18 21 20 25 24 23
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Sebastian Smyczy«ski
Generating Tree Permutations
Optimalization of the searching step
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Sebastian Smyczy«ski
Generating Tree Permutations
Optimalization of the sorting step
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9 6 3 2 1 5 4 7 8 11 10 22 19 17 13 12 16 15 14 18 21 20 25 24 23
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Sebastian Smyczy«ski
Generating Tree Permutations
Optimalization of the sorting step
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Sebastian Smyczy«ski
Generating Tree Permutations
Optimalization of the sorting step
, , , , , , , , ,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
9 6 3 2 1 5 4 7 8 11 10 22 19 17 14 12 13 15 16 18 20 21 23 24 25
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Sebastian Smyczy«ski
Generating Tree Permutations
Time Complexity
Knowing that the number of binary trees with n nodes is is given by
the Catalan number
2n
Cn =
/(n + 1).
n
and using the recursive property of tree permutations we can give
the formula for the summarized length of all working suxes of tree
permutations for given number of nodes n
n Wn = ∑ Ci − Wn−i + Wi − + (n − i )(Ci − − 1) + n(n − 1).
1
i =1
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1
Solving this recurrence we obtain Wn = Cn+ − n − 1.
Since Cn+ = (nn++ ) Cn , therefore the average-case
n
time-complexity of the algorithm is constant O ( W
Cn ) = O (1).
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Sebastian Smyczy«ski
Generating Tree Permutations
Thank you!
Binary origami tree
http://www.tsg.ne.jp/TT/origami/gallery.html
Sebastian Smyczy«ski
Generating Tree Permutations