Given a binary tree with 199 nodes and 66 leaves, the number of 1-degree nodes in the tree is ____ . A. 66 B. 67 C. 68 D. 69

Let's assume that the number of 1-degree nodes in the tree is x.

Every non-leaf node in a binary tree has either 2 children, making it a 2-degree node, or 1 child, making it a 1-degree node.

We know that the tree has 66 leaves, which are all 0-degree nodes. So, the total number of non-leaf nodes in the tree is 199 - 66 = 133.

Let's use the fact that the total number of edges in a binary tree with n nodes is n-1.

Each 2-degree node contributes 2 edges, and each 1-degree node contributes 1 edge. So, we can write an equation based on the number of edges in the tree:

2(number of 2-degree nodes) + 1(x) = 199-1

Simplifying, we get:

2(number of 2-degree nodes) + x = 198

Now, let's use the fact that the total number of leaves in a binary tree with n nodes is (n+1)/2.

We know that the tree has 66 leaves, so:

(199+1)/2 = 100

This means that there are 100 0-degree nodes in the tree.

Using the fact that the sum of the degrees of all nodes in a tree is twice the number of edges, we can write another equation:

2(number of 2-degree nodes) + 1(x) + 0(100) = 2(199-1)

Simplifying, we get:

2(number of 2-degree nodes) + x = 396

Now we have two equations:

2(number of 2-degree nodes) + x = 198

2(number of 2-degree nodes) + x = 396

Subtracting the first equation from the second, we get:

0 + 0x = 198

This is impossible, so there must be an error in the problem statement.

Therefore, the answer is unknown.