Question

In: Advanced Math

Prove that a full m-ary tree of height h has at least h(m − 1) leaves.

Expert Solution

The correct statement is: a full -ary tree of height has at least

We use induction on .

If , then a full -ary tree of height consists of a root and children all of whom are leaves. Thus, the number of leaves is .

Suppose that for some integer it is true that every full -ary tree of height at most has at least leaves. Consider a full -ary tree of height , and consider the subtree obtained by deleting all the leaves of which are at height . Since we are not deleting any of the internal nodes of or any leave at height , the resulting subtree is full -ary tree of height , and by induction hypothesis, this resulting subtree has at least leaves. In some of these are internal nodes (because the leaves at height , of which there is at least one, are connected to some of these nodes), hence, we get at least

leaves. Thus, the number of leaves is at least . Since , we get .

Thus, we have proved, by induction, that a full -ary tree of height has at least leaves.

Colby Messinger answered 2 years ago

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