Removable trees and matchings in $k$-connected and $k$-edge-connected graphs

T. Hasunuma (J. Graph Theory, 2023) conjectured that if $G$ is a $k$-connected (resp. $k$-edge-connected) graph with minimum degree $\delta(G) \ge k + m - 1$, and $T$ is a tree of order $m$, then $G$ contains a removable copy of $T$, that is, a subtree $T'$ isomorphic to $T$ such that $G - E(T')$ is $k$-connected (resp. $k$-edge-connected). We prove (a strengthening of) this conjecture. We also consider removable matchings in graphs with high minimum degree. We show, among others, that if $G$ is a $k$-edge-connected graph on at least $2m$ vertices with minimum degree $\delta(G) \ge k + m$, then there exists a matching $M$ of size $m$ in $G$ for which $G-M$ is $k$-edge-connected.