# Difference between a tree and spanning tree?!

I'm unable to understand the difference between a tree and a spanning tree. A tree is a graph that is connected and contains no circuits. A spanning tree of a graph G is a tree that contains every node of G. So what is the difference!?!

• wiki.answers.com/Q/… – dato datuashvili Feb 5 '14 at 10:33
• A tree is a type of graph. A spanning tree is a subgraph of the graph that is a tree and hits every vertex. – Ian Coley Feb 5 '14 at 10:33

"Spanning" is the difference: a spanning subgraph is a subgraph which has the same vertex set as the original graph. A spanning tree is a tree (as per the definition in the question) that is spanning.

For example:

has the spanning tree

whereas the subgraph

is not a spanning tree (it's a tree, but it's not spanning). The subgraph

is also not a spanning tree (it's spanning, but it's not a tree).

• To complete the list, you might add a spanning subgraph that is not a tree - for example the cyclic graph formed by the outer edges – Hagen von Eitzen Feb 5 '14 at 10:47
• Sure, why not. (Added.) – Rebecca J. Stones Feb 5 '14 at 10:53
• @RebeccaJ.Stones can we conclude Every Spanning tree is a tree while every tree is not necessarily a spanning tree – laura Oct 3 '17 at 6:45
• We can conclude the first part (every spanning tree is a tree) by definition. The second part sounds a bit weird. Trivially, every tree is a spanning tree of itself, but we usually think of a graph G and look at spanning trees of G (and not just think of "spanning trees" by themselves). – Rebecca J. Stones Oct 3 '17 at 7:26

A tree is just a type of graph (connected and no cycles).

You can only say that $G$ is a spanning graph of $H$: it's more of a relation between graphs, which states a few things at the same time: $G$ is a subgraph of $H$ (i.e. it has a subset of the vertices and a subset of the edges), $G$ is a tree when considered on its own (as above), and it is spanning (the set of vertices of $G$ actually equals the vertices of $H$). So it says three things, of which two are about the relation between them. Saying it is a tree is simpler and has less information.

The only difference is the word 'spanning', a kind of 'skeleton' which is just capable to hold the structure of the given graph $G$. Infact, there may be more than one such 'skeletons' in a given graph but a tree $T$ has the only one i.e. $T$ itself.

Spanning tree is a maximal tree subgraph or maximal tree of graph G (i.e. A tree T is said to be a spanning tree of a connected graph G if T is a subgraph of G and T contain all vertices of G.