$1$-page oriented book embedding guaranteed by Theorem \[oriented tree\]. To obtain a $1$-page oriented book embedding of $\overrightarrow{T}$, embed the vertices into the spine with spine order $(x, \beta_1, \beta_2,...,\beta_k)$ such that $x$ is the top vertex in the spine. By Theorem \[oriented tree\], every arc of $\overrightarrow{T_{v_i}}$, $1 \le i \le k$, can be placed into the interior of the page such that $v_i$ is uncovered, each arc is upwards, and each vertex is loose. Then place each arc $(v_i,x)$ into the interior of the page, and since all arcs $(v_i,x)$ share $x$ as a common endpoint, they are nested and thus do not cross in the page. Therefore the statement holds.

Let $\overrightarrow{T}$ be an oriented tree with sink $x$. For convenience, we call an oriented book embedding, as described in Lemma \[tree\_sink\], a *sink oriented book embedding*; we abbreviate the spine order of such an embedding $(x;\alpha_x)$, such that $x$ is the top vertex in the spine, and $\alpha_x$ represents $( \beta_1, \beta_2,...,\beta_k)$. Similarly, we can construct a *source oriented book embedding*.

We are now prepared to define a sink fountain tree having an specific $1$-page oriented book embedding, which will be useful in Section 4. To 