TAOCP 2.3 Exercise 4
False.
Exercise 4. [**] [01] True or false: In a conventional tree diagram (root at the top), if node X has a higher level number than node Y then node X appears lower in the diagram than node Y.
Verified: no
Solve time: 35s
False.
The level of a node is determined by its distance from the root, but a conventional tree diagram does not require all nodes of the same level to lie on a common horizontal line, nor does it require every node of level $k+1$ to be drawn lower than every node of level $k$. The definitions of a tree and of level numbers impose no geometric restrictions on the drawing.
A counterexample is obtained by drawing a node $X$ of level $2$ slightly above a node $Y$ of level $1$ while preserving the parent-child connections. Then $X$ has a higher level number than $Y$, yet $X$ appears higher in the diagram. Therefore the statement is false.
$$ \boxed{\text{False}} $$