Pair leaves by edge-disjoint paths and find edge-disjoint cycles through a vertex

Prove that a tree having \(2k\) endvertices contains \(k\) edge-disjoint paths that pair all of its endvertices.

Deduce that if \(x\) is a vertex of a graph \(G\) with degree \(2k\) and \(x\) is not a cutvertex of \(G\), then \(x\) lies on \(k\) edge-disjoint cycles.