COMBINATORICS. PROBLEM SET 10.
ENUMERATION OF TREES
SolrruaR PRoBLEMS
Problem 10.1. Find the nurnber of non-directed graphs on ??vertices
with simple edges and no ioops.
Problem
vertices?
1O.2. What is the number of rooted plane trees with n -t 1
Problem
10.3. Consider the following three triangulations of the hexagorr:
'. -''\..
.,'l-.-'
' ' \
i
\i
L
.1
n.."
'L1-"rr'
/.
..
>
.,,'
Assumc we are rotating the hexagon by 60o, 720", .. . . How ntanv other
different triangulations each picture produces?
Problem LO.4. How many triangulationsof the (n * 2)-gon with labclcd
verticesare? What about the (n * 2)-gon with unlabeledvertices? Let
the latter number of triangulationsbe Rr,. Give estimatesfor R, of the
forrn a,, 1 R, ( b,-,,such Lhal bnfa, has a polynomial asymptotics.
Problem 10.5. Give a recursivedefinition of
x) a rootcd tree
x) a labeledrooted tree
x) a plane tree
*) a labeledplane tree
Problem 10.6 (Cavley's formula). Lel Tn be the number of labeled
rootcd trccson n verticcs,and 7(r) :: D,,>r Tnr" f nl. Showthat 7(r) :
,"1'('). Applying the Lagrange inversion,shorvthat 4 : 77't-r.
,[
\ti I
-
Problem LO.7. Computc the Priifcr codc of thc following trcc:
--{E\ -,
(t----i.t)
'g
\7
\
1,..
\
(+j\
\
\ - , \ ; - \ r -' .,--/- -
\
\n,
(3,'
(t;_ri-)
| ,-- . _
\_j
_!,
i
/\---I \ , ,' ,( -1-,r t a ' : '
Problem 10.8. Constructthe labeFflur.root'.'d{r."froriit, Priifer code
( 3 ,1 , 5 , 3 , 5 . 3 , 12 ,1 0 ) .
Problem 10.9. Obtain Problem 10.6using Pnifer cocles.
Problem 10.10. From Problem 10.6 deducethe nuntber of uttrooted
labeledtrees on n vertices.
Problem 10.11.
Problem 10.12. Generalizethe consructionof a Priifer code for labeled
rooted forests.
2
CONIBINATORICS. PROBLE\,I SET 10. trNUMERATION OF TREES
HoltnwoRx
Problem 10.13 (1). Find thc number of non-dircctcdgraphs on ?zvcrtices with simple edges(loops allowed).
Problem 10.14 (1). Find the number of directedgraphson n vcrtices
rn'ithsinrpleedges(loopsallowed).
Problem 10. 15 ( 7 ) . What is the number of rooted plane treesrvith n f 2
verticessuch that the degreeof the root is one?
Problem 10.16 (1). How many differentcyclic orclersare on n elerrtents'/
( c . g . ,f o r n : 3 t h c r c a r c t w o s u c ho r d c r s :1 2 3a n d 1 3 2 ) .
Problem 10.17 (1). Draw all possible(unlabeled,unrooted)treeson n
v e r t i c e sn, : 1 ,2 , 3 , 4 . 5 , 6 .
Problem 10.18 (2). Compute the Priifer code of the following trees:
-g
,)
"') i.1
"
> ) I,i,/' IY
'; 5o---*-iu tT:= t
\rhr;,
:7
I o!- i . . -oL'2
fHf i H
Lt
,
.l u-_*-*l *+!7-**-
sr=-€'<|-o)-ff;j
-,f::
Problem 10.19. Construct the labeledunrooted trees frorn their Prr-ifer
c o d e su: ) ( 7 ) ( 1 , 2 , 3 , 4 , 5 , 6 , 1 , 2 ,b3))(, 7 ) ( 3 , 5 , 3 , 5 , 3 , 5 , 3 , 5 ) .
Problem 10.20 (3). Use the Priif'ercode constructionto show that the
number of labeled unrooted trees on n verticeswith degreesof vertices
(n - 2\l
dt,,..,rl,.(cliisthedegreeofthe7thvertex)i'ffi
can occur from such tlees.)
(Hint: think of what Priifer sequences
Problem LO.2L (3). Consider biparti'tetrees (labeledunrooted), i.e.'
treesin which verticesare of two types: black verticeswith labelsI, . . . , flb,
and white verticeswith labelsnt iI,. .. ,Ttb*T1,,.Each edgemust connect
a white and a black vertex. Show that the number of such bipartite trees
is nff-,rn["'-1. (Argue similarly to the previonsproblem.)
Problem LO.22 (3). Take a rooted labeled forest on n vertices. Take
roots of each component and connect them to a new vertex with label
n-t-1. Observethat thiswill be an unrootedlabelcdtree on nf l vertir:cs.
Deducethat the numberof rootedlabeledforestson n verticesis (n,*1)''-1.
PRoBLEM
SuppI-BI,IBNTARY
Problem 10.23 (5). Deducethat the number of labeled rooted forests
o1 n vcrticcs with k componcntssuch that the root of thc zth componcnt
has label i.. ts kn"-k-l (useProblem10.12).