How many distinct positional 3-trees are there with height 2?
To Find how many distinct positional 3-trees, we can follow these three steps.
1. With one link, from level 0 to level 1, there are 3 possible ways:
0
2
1
1
2
0
3
1
2
3
3
1
2
1
3
1
2
0
3
2
3
1
2
3
1
1
2
3
1
2
2
3
1
2
3
3
1
2
3
3
For each of these three ways we can link the level 1 with level 2 in 2 − 1 = 7 ways. We can understand this formula as follows. There
are 2 possible ways: either to link the level 1 with vertex i or not. Because there are three vertices in level 2. Then the total of ways
is 23 . We have to subtract from this number the case where there is no link ( there is only once case). So we get 23 − 1.
Therefore there are 3 × 7 = 21.
2. With two links, from level 0 to level 1, there are also 3 possible ways:
0
1
1
2
0
3
2
3
1
2
3
1
2
1
3
1
2
0
2
3
1
2
3
3
1
2
1
2
3
1
6
2
3
1
2
3
3
1
2
3
For each of these three ways we can link the level 1 with level 2 in 2 − 1 = 63 ways. We can understand this formula as follows.
There are 2 possible ways: either to link the level 1 with vertex i or not. Because there are six vertices in level 2. Then the total of
ways is 26 . We have to subtract from this number the case where there is no link ( there is only once case). So we get 26 − 1.
Therefore there are 3 × 63 = 189.
3. With three links, from level 0 to level 1, there is only one way:
0
1
1
2
2
3
1
2
3
3
1
2
3
In a similar way, the number of ways to link level 1 with level 2 is given by 29 − 1 = 511.
Thus, there are 21 + 189 + 511 = 721 different positional 3-trees.