* Structural Induction
* Other Methods of Proof for Trees
* Node Induction
In 103A, we saw a number of inductive proofs concerning integers and sequences.
assume some statement for n, or for all integers less than or equal to n (strong induction), and use
this "inductive hypothesis" to prove the same statement for n+1.
A similar form of proof is useful
for concluding facts about trees.
Suppose we want to prove that a statement S(T) is true for all trees.
For a basis, we show that S(T) is
true whenever T consists of one node.
For the induction, we suppose that T is a tree with root r and
children c1, c2, .
Let T1, T2, .
.., Tk be the subtrees of T rooted at c1, c2, .
.., ck respectively as
The inductive step is to assume S(T1), S(T2), .
.., S(Tk) are all true, and prove S(T).
If we can do
this, we have proven S(T).
Why does structural induction work?
S(T) denotes that there are at most m
leaves in an m-ary tree of height h.
root is at level 1 with a height of 0)
Consider an m-ary tree of height 1.
These trees consist of a root with no more than m
children, each of which is a leaf.
Therefore, there are no more than m
= m leaves in an m-
ary tree of height 1.
: We assume the above statement is true for all m-ary trees of height less
than h, and show it is true for all m-ary trees of height h.
Let T be an m-ary tree of height h.
The leaves of T are the leaves of the subtrees of T
obtained by deleting the edges from the root to each of the vertices connected to it, as shown