Let $G$ be a connected graph with vertex set $V(G)=\{v_1,v_2,\dots,v_n\}$ and edge set $E(G)$. $D(G)=(d_{ij})_{n\times n}$ is the distance matrix of $G$, where $d_{ij}$ denotes the distance between $v_i$ and $v_j$. Let $\lambda_1(D)\geq\lambda_2(D)\geq\dots\geq\lambda_n(D)$ be the distance spectrum of $G$. A graph $G$ is said to be determined by its distance spectrum if any graph having the same distance spectrum as $G$ is isomorphic to $G$. Trees can not be determined by its distance spectrum. Naturally, we prove that two kinds of special trees path $P_n$ and double star $S(a,b)$ are determined by their distance spectra in this paper.