  We apply the Euler tour technique to find subtrees of specified weight as
follows. Let $k, g, N_1, N_2 \in \mathbb{N}$ such that $1 \leq k \leq N_2$, $g
+ h > 2$ and $2k - 4g - h + 3 \leq N_2 \leq 2k + g + h - 2$, where $h := 2N_1 -
N_2$. Let $T$ be a tree of $N_1$ vertices and let $c : V(T) \rightarrow
\mathbb{N}$ be vertex weights such that $c(T) := \sum_{v \in V(T)} c(v) = N_2$
and $c(v) \leq k$ for all $v \in V(T)$. We prove that a subtree $S$ of $T$ of
weight $k - g + 1 \leq c(S) \leq k$ exists and can be found in linear time. We
apply it to show, among others, the following: (i) Every planar hamiltonian
graph $G = (V(G), E(G))$ with minimum degree $\delta \geq 4$ has a cycle of
length $k$ for every $k \in \{\lfloor \frac{|V(G)|}{2} \rfloor, \dots, \lceil
\frac{|V(G)|}{2} \rceil + 3\}$ with $3 \leq k \leq |V(G)|$. (ii) Every
$3$-connected planar hamiltonian graph $G$ with $\delta \geq 4$ and $|V(G)|
\geq 8$ even has a cycle of length $\frac{|V(G)|}{2} - 1$ or $\frac{|V(G)|}{2}
- 2$. Each of these cycles can be found in linear time if a Hamilton cycle of
the graph is given. This work was partially motivated by conjectures of Bondy
and Malkevitch on cycle spectra of 4-connected planar graphs.
