Bijective proofs for Eulerian numbers of types B and D

Let $\Bigl\langle\matrix{n\cr k}\Bigr\rangle$, $\Bigl\langle\matrix{Bn\cr k}\Bigr\rangle$, and $\Bigl\langle\matrix{Dn\cr k}\Bigr\rangle$ be the Eulerian numbers in the types A, B, and D, respectively -- that is, the number of permutations of n elements with $k$ descents, the number of signed permutations (of $n$ elements) with $k$ type B descents, the number of even signed permutations (of $n$ elements) with $k$ type D descents. Let $Sn(t) = \sum{k = 0}^{n-1} \Bigl\langle\matrix{n\cr k}\Bigr\rangle t^k$, $Bn(t) = \sum{k = 0}ⁿ \Bigl\langle\matrix{Bn\cr k}\Bigr\rangle t^k$, and $Dn(t) = \sum{k = 0}ⁿ \Bigl\langle\matrix{Dn\cr k}\Bigr\rangle t^k$. We give bijective proofs of the identity $$Bn(t²⁾ = (1 + t)^{n+1}Sn(t) - 2ⁿ tSn(t^2)$$ and of Stembridge's identity $$Dn(t) = Bn(t) - n2^{n-1}tS{n-1}(t).$$ These bijective proofs rely on a representation of signed permutations as paths. Using this representation we also establish a bijective correspondence between even signed permutations and pairs $(w, E)$ with $([n], E)$ a threshold graph and $w$ a degree ordering of $([n], E)$, which we use to obtain bijective proofs of enumerative results for threshold graphs.