The main purpose of this paper is to explore the structure of regular subspaces of 1-dim Brownian motion.As outlined in /cite{FMG} every such regular subspace can be characterized by a measure-dense set $G$.When $G$ is open, $F=G^c$ is the boundary of $G$ and, before leaving $G$, the diffusion associated with the regular subspace is nothing but Brownian motion. Their traces on $F$ still inherit the inclusion relation,in other words, the trace Dirichlet form of regular subspace on $F$ is stall a regular subspace of trace Dirichlet form of one-dimensional Brownian motion on $F$.Moreover we have proved that the trace of Brownian motion on $F$ may be decomposed into two part, one is the trace of the regular subspace on $F$, which has only the non-local part and the other comes from the orthogonal complement of the regular subspace, which has only the local part. Actually %the former one is a non-local Dirichlet form whereas the latter one has non-trivial local part. The remaining information, i.e. the information of strongly local part of trace Dirichlet form of one-dimensional Brownian motion on $F$, is contained in the orthogonal complement of regular subspace corresponds to a time-changed Brownian motion after a darning transform.