Bridge Tree Example

Graph: $0-1$, $1-2$, $2-0$, $1-3$, $3-4$, $4-3$. $2$-edge-connected components: $\{0,1,2\}$ (cycle) and $\{3,4\}$ (cycle). Bridge: $1-3$. Bridge tree: node $A = \{0,1,2\}$, node $B = \{3,4\}$, edge $A-B$. The bridge tree has two nodes connected by one edge.

You can now solve tree problems (like LCA, diameter, paths) on this compressed structure instead of the original graph. This simplifies many algorithms.