# Graph theory and trees

Let T be a tree with n vertices,where n greater than or equal to 3.Show that there is a vertex V in T with d(V) greater than or equal to 2 such that every vertex adjacent to V ,except possibly for one ,has degree 1.

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Let $$T’$$ be a subgraph of $$T$$ induced by the set of all vertices of $$T$$ which have degree at least $$2$$. Since $$T$$ is connected and $$n\ge 3$$, $$T’$$ is non-empty. Since $$T’$$ is a subgraph of a tree, it is forest, that is a union of mutually disjoint trees. Choose any of these trees and pick as the required vertex $$v$$ any its leaf, that is a vertex of degree $$1$$.