  A birth and death process is a continuous-time Markov chain with the minimal
state space $\mathbb N$, whose transition matrix is standard and whose density
matrix is the given birth-death matrix. Birth and death process is unique if
and only if $\infty$ is an entrance or natural. When $\infty$ is neither an
entrance nor natural, there are two ways in the literature to obtain all birth
and death processes. The first one is an analytic treatment proposed by Feller
in 1959, and the second one is a probabilistic construction completed by Wang
in 1958.
  In this paper we will give another way to study birth and death processes
using the Ray-Knight compactification. This way has the advantage of both the
analytic and probabilistic treatments above. By applying the Ray-Knight
compactification, every birth and death process can be modified into a
c\`adl\`ag Ray process on $\mathbb N\cup \{\infty\}\cup\{\partial\}$, which is
either a Doob processes or a Feller $Q$-process. Every birth and death process
in the second class has a modification that is a Feller process on $\mathbb
N\cup\{\infty\}\cup \{\partial\}$. We will derive the expression of its
infinitesimal generator, which explains its boundary behaviours at $\infty$.
Furthermore, by utilizing transformations of killing and
Ikeda-Nagasawa-Watanabe's piecing out, we will also provide a probabilistic
construction for birth and death processes. This construction relies on a
triple determining the resolvent matrix introduced by Wang and Yang in their
work (1992')}.
