  A permutation $\sigma\in\mathfrak{S}_n$ is simsun if for all $k$, the subword
of $\sigma$ restricted to $\{1,...,k\}$ does not have three consecutive
decreasing elements. The permutation $\sigma$ is double simsun if both $\sigma$
and $\sigma^{-1}$ are simsun. In this paper we present a new bijection between
simsun permutations and increasing 1-2 trees, and show a number of interesting
consequences of this bijection in the enumeration of pattern-avoiding simsun
and double simsun permutations. We also enumerate the double simsun
permutations that avoid each pattern of length three.
