We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G=(V,E)$ and a subset $A$ of $E$ we let ${\lambda }_G\left(A\right)$ be the number of vertices incident with an edge in $A$ and an edge in $E\setminus A$. For a subset $X$ of $V$, let ${\rho }_G\left(X\right)$ be the rank of the adjacency matrix between $X$ and $V\setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions ${\lambda }_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions ${\rho }_G$ has bounded branch-depth, which we call the rank-depth of graphs.

Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by restriction.

我们提出了一个称为连通性功能的分支深度的概念，该概念概括了图的树深度。然后，我们证明了两个定理，这些定理表明该概念与图的树深度和灌木深度概念紧密一致。对于图$G=（V，Ë）$ 和一个子集 $\mathrm{一种}$ 的 $Ë$ 我们让 ${\lambda }_G（\mathrm{一种}）$ 是入射边中的顶点的数量 $\mathrm{一种}$ 和边缘 $Ë\setminus \mathrm{一种}$。对于子集$X$ 的 $V$，让 ${\rho }_G（X）$ 是之间的邻接矩阵的等级 $X$ 和 $V\setminus X$在二进制字段上。我们证明，当且仅当对应的函数类别时，一类图才具有边界树深度${\lambda }_G$ 当且仅当相应的函数类具有边界分支深度，并且类似地，一类图具有边界灌木深度 ${\rho }_G$ 限制了分支深度，我们称其为图的秩深度。

此外，我们研究了树深对拟阵的各种潜在概括，并证明了在没有大电路的固定有限域上可表示的拟阵受到限制是准排序的。