Prove the following recurrence.
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A full binary tree is a tree in which every node other than the leaves has two children. Let $K_n$ be the number of full binary trees with $n+1$ leaves. Show that
$$K_n = sum_{i=0}^{n-1}K_{i}K_{n-1-i}.$$
I thought of splitting the tree by its root node into sub-trees with maybe $i$ and $n-i$ nodes, but I am not sure how to prove this recurrence formally. Any ideas?
recurrence-relations
asked Nov 15 at 19:43
Hello_World
3,77521630
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up vote
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A full binary tree is a tree in which every node other than the leaves has two children. Let $K_n$ be the number of full binary trees with $n+1$ leaves. Show that
$$K_n = sum_{i=0}^{n-1}K_{i}K_{n-1-i}.$$
I thought of splitting the tree by its root node into sub-trees with maybe $i$ and $n-i$ nodes, but I am not sure how to prove this recurrence formally. Any ideas?
recurrence-relations
asked Nov 15 at 19:43
Hello_World
3,77521630
Use induction. First show that the formula is true for $n=1$. Then induct: prove that if the formula is true for $n$, then it's true for $n+1$.
– Nick
Nov 16 at 17:53
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
A full binary tree is a tree in which every node other than the leaves has two children. Let $K_n$ be the number of full binary trees with $n+1$ leaves. Show that
$$K_n = sum_{i=0}^{n-1}K_{i}K_{n-1-i}.$$
I thought of splitting the tree by its root node into sub-trees with maybe $i$ and $n-i$ nodes, but I am not sure how to prove this recurrence formally. Any ideas?
recurrence-relations
asked Nov 15 at 19:43
Hello_World
3,77521630
A full binary tree is a tree in which every node other than the leaves has two children. Let $K_n$ be the number of full binary trees with $n+1$ leaves. Show that
$$K_n = sum_{i=0}^{n-1}K_{i}K_{n-1-i}.$$
I thought of splitting the tree by its root node into sub-trees with maybe $i$ and $n-i$ nodes, but I am not sure how to prove this recurrence formally. Any ideas?
recurrence-relations
recurrence-relations
asked Nov 15 at 19:43
Hello_World
3,77521630
asked Nov 15 at 19:43
Hello_World
3,77521630
edited Nov 16 at 17:04
asked Nov 15 at 19:43
Hello_World
3,77521630
asked Nov 15 at 19:43
Hello_World
3,77521630
asked Nov 15 at 19:43
Hello_World
3,77521630
3,77521630
Use induction. First show that the formula is true for $n=1$. Then induct: prove that if the formula is true for $n$, then it's true for $n+1$.
– Nick
Nov 16 at 17:53
add a comment |
Use induction. First show that the formula is true for $n=1$. Then induct: prove that if the formula is true for $n$, then it's true for $n+1$.
– Nick
Nov 16 at 17:53
Use induction. First show that the formula is true for $n=1$. Then induct: prove that if the formula is true for $n$, then it's true for $n+1$.
– Nick
Nov 16 at 17:53
Use induction. First show that the formula is true for $n=1$. Then induct: prove that if the formula is true for $n$, then it's true for $n+1$.
– Nick
Nov 16 at 17:53
add a comment |
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Use induction. First show that the formula is true for $n=1$. Then induct: prove that if the formula is true for $n$, then it's true for $n+1$.
– Nick
Nov 16 at 17:53