What is the total number of nodes in a full k-ary tree, in terms of the number of leaves?
Think about how to prove the result for a full binary tree, and you'll see how to do it in general. For the full binary tree, say of height h
, the number of nodes N
is
N = 2^{h+1} - 1
Why? Because the first level has 2^0
nodes, the second level has 2^1
nodes, and, in general, the k
th level has 2^{k-1}
nodes. Adding these up for a total of h+1
levels (so height h
) gives
N = 1 + 2 + 2^2 + 2^3 + ... + 2^h = (2^{h+1} - 1) / (2 - 1) = 2^{h+1} - 1
The total number of leaves L
is just the number of nodes at the last level, so L = 2^h
. Therefore, by substitution, we get
N = 2*L - 1
For a k
-ary tree, nothing changes but the 2
. So
N = 1 + k + k^2 + k^3 + ... + k^h = (k^{h+1} - 1) / (k - 1)
L = k^h
and so a bit of algebra can take you the final step to get
N = (k*L - 1) / (k-1)
Andrew
Updated on November 17, 2021Comments
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Andrew over 2 years
I am doing a unique form of Huffman encoding, and am constructing a k-ary (in this particular case, 3-ary) tree that is full (every node will have 0 or k children), and I know how many leaves it will have before I construct it. How do I calculate the total number of nodes in the tree in terms of the number of leaves?
I know that in the case of a full binary tree (2-ary), the formula for this is 2L - 1, where L is the number of leaves. I would like to extend this principle to the case of a k-ary tree.
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BlackSwan over 6 years[(k^(h+1))-1]/(h-1) is wrong. It's [(k^(h+1))-1]/(k-1). Please verify answers before posting.