Question

let be a tree having vertices. suppose that every vertex in which is not a leaf has degree , where . find a formula, in terms of and , for the number of leaves of .

Answer 1

The formula for the number of leaves in a tree in graph theory, with n vertices where each non-leaf vertex has degree k, is l = 2 + (k-2)n.

Step-by-step explanation:

The question asks us to find a formula for the number of leaves of a tree in graph theory. Assuming there are n vertices and every vertex that is not a leaf has a degree of k, where k > 1.

In graph theory, the number of leaves (l) can be found using the formula derived from the Handshaking Lemma; the sum of degrees of all vertices equals twice the number of edges.

Let's say e is the number of edges. Then, l + k(n-l) = 2e. Since a tree with n vertices has n-1 edges, we substitute e with n-1 to obtain l + k(n-l) = 2(n-1). After rearranging, we get l = 2 + (k-2)n.