Final Answer:
The given statement " Every edge in a tree is a cut-edge" is False. While every edge in a tree is crucial for maintaining its connectivity, not every edge is a cut-edge.
Step-by-step explanation:
In a tree, a cut-edge refers to an edge that, upon removal, increases the number of connected components. Trees have no cycles and are connected, meaning each edge contributes to maintaining this connectivity. However, a cut-edge specifically refers to an edge that, when removed, separates the tree into two or more distinct parts.
Consider a tree with multiple branches connected at a single node. Each edge, though vital for the tree's structure, may not necessarily be a cut-edge. Cutting an edge from such a tree might disconnect a branch but won't split the entire tree into separate components. Therefore, while all edges in a tree are crucial for its connectivity, not all edges are cut-edges.
In graph theory, distinguishing between edges that are fundamental for connectivity (tree edges) and those whose removal disconnects the graph (cut-edges) helps in understanding the structural properties of trees and graphs. Trees possess specific characteristics where every edge contributes to the tree's connectedness, but only specific edges serve as cut-edges, altering its connected components upon removal.