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Let $\mathbb{G}^{D}$ be the set of graphs $G(V,\, E)$ with$\left|V\right|=n$, and the degree sequence equal to $D=(d_{1},\,d_{2},\,\dots,\, d_{n})$. In addition, for $\frac{1}{2}<a<1$, we define the setof graphs with an almost given degree sequence $D$ as follows, \[\mathbb{G}_{a}^{D}:=\cup\,\mathbb{G}^{\bar{D}}, \] where the union is over alldegree sequences $\bar{D}$ such that, for $1\leq i\leq n$, we have$\left|d_{i}-\bar{d}_{i}\right|<d_{i}^{a}$.  Now, if we chose random graphs $\mathcal{G}_{\mathbf{g}}\left(D\right)$ and$\mathcal{G}_{\mathbf{a}}\left(D\right)$ uniformly out of the sets$\mathbb{G}^{D}$ and $\mathbb{G}_{a}^{D}$, respectively, what do they looklike? This has been studied when $\mathcal{G}_{\mathbf{g}}\left(D\right)$ is adense graph, i.e. $\left|E\right|=\Theta(n^{2})$, in the sense of graphons, orwhen $\mathcal{G}_{\mathbf{g}}\left(D\right)$ is very sparse, i.e.$d_{n}^{2}=o(\left|E\right|)$. In the case of sparse graphs with an almostgiven degree sequence, we investigate this question, and give the finite treesubgraph structure of $\mathcal{G}_{\mathbf{a}}\left(D\right)$ under some mildconditions. For the random graph $\mathcal{G}_{\mathbf{g}}\left(D\right)$ witha given degree sequence, we re-derive the finite tree structure in dense andvery sparse cases to give a continuous picture.  Moreover, for a pair of vectors$\left(D_{1},D_{2}\right)\in\mathbb{Z}^{n_{1}}\times\mathbb{Z}^{n_{2}}$, we let$\mathcal{G}_{\mathbf{b}}\left(D_{1},D_{2}\right)$ be the random bipartitegraph that is chosen uniformly out of the set $\mathbb{G}^{D_{1},D_{2}}$, where$\mathbb{G}^{D_{1},D_{2}}$ is the set of all bipartite graphs with the degreesequence $\left(D_{1},D_{2}\right)$. We are able to show the result for$\mathcal{G}_{\mathbf{b}}\left(D_{1},D_{2}\right)$ without any furtherconditions.

Trees in Random Sparse Graphs with a Given Degree Sequence