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A binary tree can be...

A binary tree can be recursively defined as follows.

A single vertex is a binary tree. (This single vertex is the root of this binary tree.)

If T1 and T2 are binary trees, then so is the result of the following construction.

– Create a new vertex. (This vertex will become the root of the new binary tree.)

– Add an edge between this new root vertex and the roots of T1 and T2.

Prove the following for all binary trees:

• Root vertices either have degree 0 or 2 and are the only vertices with these degrees.

The number of degree 1 vertices is 2 more than the number of degree 3 vertices.

(Hint: It will be simpler to combine the two properties in a single proof.)

A single vertex is a binary tree. (This single vertex is the root of this binary tree.)

If T1 and T2 are binary trees, then so is the result of the following construction.

– Create a new vertex. (This vertex will become the root of the new binary tree.)

– Add an edge between this new root vertex and the roots of T1 and T2.

Prove the following for all binary trees:

• Root vertices either have degree 0 or 2 and are the only vertices with these degrees.

The number of degree 1 vertices is 2 more than the number of degree 3 vertices.

(Hint: It will be simpler to combine the two properties in a single proof.)

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