----- > [!proposition] Proposition. ([[average degree of tree is strictly less than two]]) > The average node [[degree]] $c$ of a [[tree]] $T$ is strictly less than two. > [!proof]- Proof. ([[average degree of tree is strictly less than two]]) > Suppose $T$ has $n$ nodes. Then $T$ has $n-1$ edges by [[tree iff has n-1 edges]]. The average node [[degree]] of any [[graph]] is equal to $\frac{2m}{n}$, where $m$ is the number of edges in said [[graph]]. For the case of $T$, this implies $c=\frac{2m}{n}=\frac{2(n-1)}{n} < 2.$ ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```