  In the binomial random graph $\mathcal{G}(n,p)$, when $p$ changes from
$(1-\varepsilon)/n$ (subcritical case) to $1/n$ and then to $(1+\varepsilon)/n$
(supercritical case) for $\varepsilon>0$, with high probability the order of
the largest component increases smoothly from
$O(\varepsilon^{-2}\log(\varepsilon^3 n))$ to $\Theta(n^{2/3})$ and then to $(1
\pm o(1)) 2 \varepsilon n$.
  As a natural generalisation of random graphs and connectedness, we consider
the binomial random $k$-uniform hypergraph $\mathcal{H}^k(n,p)$ (where each
$k$-tuple of vertices is present as a hyperedge with probability $p$
independently) and the following notion of high-order connectedness. Given an
integer $1 \leq j \leq k-1$, two sets of $j$ vertices are called
\emph{$j$-connected} if there is a walk of hyperedges between them such that
any two consecutive hyperedges intersect in at least $j$ vertices. A
$j$-connected component is a maximal collection of pairwise $j$-connected
$j$-tuples of vertices. Recently, the threshold for the appearance of the giant
$j$-connected component in $\mathcal{H}^k(n,p)$ and its order were determined.
In this article, we take a closer look at the subcritical random hypergraph. We
determine the structure, order, and size of the largest $j$-connected
components, with the help of a certain class of `hypertrees' and related
objects. In our proofs, we combine various probabilistic and enumerative
techniques, such as generating functions and couplings with branching
processes. Our study will pave the way to establishing a symmetry between the
subcritical random hypergraph and the hypergraph obtained from the
supercritical random hypergraph by deleting its giant $j$-connected component.
