tree

a connected acyclic graph is called a tree (hence a tree is a connected forest).
[cite:;taken from @graph_diestel_2017 chapter 1.5 trees and forests]

the following cases are equivalent:

the graph $\text{[math]}$ is a tree,
between every two vertices there is a single path,
$\text{[math]}$ is connected and upon the removal of any of the edges we get a non-connected graph,
$\text{[math]}$ is connected and $\text{[math]}$ ,
$\text{[math]}$ is acyclic and $\text{[math]}$ .

the degree of a tree is a the highest degree that any of its nodes have.

the height of a tree (sometimes referred to as depth of a tree) is the number of the vertices in the longest path downwards (including the root node).

the depth of a vertex is the number of edges from the root to it.

the height of a vertex in a (rooted) tree is the length of the longest downward path to a leaf from that vertex.

the level of a row of vertices or a vertex in a (rooted) tree is the depth of one of the vertices (a row contains vertices all at the same depth).

$\text{[math]}$