Trees and Spanning Trees A tree is a connected graph G that is structurally equivalent to a single, connected planar graph. In simpler words, it is a co...
Trees and Spanning Trees
A tree is a connected graph G that is structurally equivalent to a single, connected planar graph. In simpler words, it is a connected graph that can be drawn on a sheet of paper without lifting the pen from the paper.
A tree can have multiple components, but it will always have at least two components: a root node and one or more branches. A spanning tree is a subset of the edges of a graph that connects all of the nodes in the graph. In other words, a spanning tree is a subgraph that can be used to cover the entire graph without leaving any nodes uncovered.
Trees and spanning trees are both fundamental concepts in graph theory. They are used to model real-world systems, such as social networks, transportation networks, and electrical circuits.
Examples:
A simple tree is a graph with three nodes and four edges, where the edges connect the following pairs of nodes:
Node 1 to node 2
Node 1 to node 3
Node 2 to node 4
A spanning tree for this graph is the following subgraph:
{node 1, node 2, node 3}
{node 1, node 2, node 4}
Additional Notes:
A graph G is connected if there is a path between any two nodes in the graph.
A spanning tree for a graph G is a subset of the edges of G that connects all of the nodes in G.
A tree is connected, but a spanning tree is not.
The degree of a node in a tree is the number of edges that connect to that node.
The depth of a node in a tree is the number of edges from the root node to that node