  The Strong Nine Dragon Tree Conjecture asserts that for any integers $k$ and
$d$ any graph with fractional arboricity at most $k + \frac{d}{d+k+1}$
decomposes into $k+1$ forests, such that for at least one of the forests, every
connected component contains at most $d$ edges. We prove this conjecture when
$d \leq k+1$.
  We also prove an approximate version of this conjecture, that is, we prove
that for any positive integers $k$ and $d$, any graph with fractional
arboricity at most $k + \frac{d}{d+k+1}$ decomposes into $k+1$ forests, such
that one for at least one of the forests, every connected component contains at
most $d + \frac{d(k (2\lceil \frac{d}{k+1} +2 \rceil)^{\lceil \frac{d}{k+1} +
2) \rceil} - k)}{k+1} $ edges.
