Let λ 1 ( T ) and λ 2 ( T ) be the largest and the second largest eigenvalues of a tree T , respectively. We obtain the following sharp lower bound for λ 1 ( T ) : λ 1 ( T ) ≥ max { d i + m i − 1 } , where d i is the degree of the vertex v i and m i is the average degree of the adjacent vertices of v i . Equality holds if and only if T is a tree T ( d i , d j ) , where T ( d i , d j ) is formed by joining the centers of d i copies of K 1 , d j − 1 to a new vertex v i , that is T ( d i , d j ) − v i = d i K 1 , d j − 1 . Let d 1 and d 2 be the highest and the second highest degree of T , respectively. Let r ( T ) be the maximum distance between the highest and the second highest degree vertices. We also show that if T is a tree of order ( n > 2), then λ 2 ( T ) ≥ { d 1 + d 2 − 1 − ( d 1 + d 2 − 1 ) 2 − 4 ( d 1 − 1 ) ( d 2 − 1 ) 2 if r ( T ) = 1 , d 1 + d 2 − ( d 1 − d 2 ) 2 + 4 2 if r ( T ) = 2 , d 1 − 1 if r ( T ) = 3 and d 1 = d 2 , d 2 otherwise . The equality holds if T is a tree T 1 or a tree T 2 , or T is a tree T 4 and d 1 = d 2 , where T 1 is formed by joining the centers of K 1 , d 1 − 1 and K 1 , d 2 − 1 and T 2 is formed by joining the centers of K 1 , d 1 − 1 and K 1 , d 2 − 1 to a new vertex, the T 4 is formed by joining a 1-degree vertex of K 1 , d 1 and K 1 , d 2 to a new vertex.