applied to any infinity point of the form $(a:b:0),\,a\not=0$. In the following, we assume that $a=0$; that is, we take the infinity point $P=(0:1:0)$. In this case, we consider the curve defined implicitly by the polynomial $h(x,z)=F(x:1:z)$. Observe that $h(p)=0,$ where $p=(0,0)$. In this situation, we get that there exists $M \in {\Bbb R}^+$ such that $$F(\varphi(t):1:t)=h(\varphi(t), t)=0,\quad \mbox{for\, $t\in {\Bbb C}$\, and
$|t|<M$,\,\,\quad where}$$ $$\varphi(z)=a_1z^{N_1/N}+a_2z^{N_2/N}+a_3z^{N_3/N} + \cdots\in{\Bbb C}\ll z\gg,\quad a_i\not=0,\, \forall i\in {\Bbb N}$$ $N,N_i\in {\Bbb N},\,\,i=1,\ldots$, and $0<N_1<N_2<\cdots$, is a series expansion for a solution of $h(x,z)=0$. We set $z=t^{-1}$, and we get that $$f(r(z),z)=0,\quad \mbox{$z\in {\Bbb C}$\, and
$|z|>M^{-1}$,\qquad where}$$ $$r(z)=z\varphi(z^{-1})=a_1z^{1-N_1/N}+a_2z^{1-N_2/N}+a_3z^{1-N_3/N} + \cdots,\quad a_i\not=0,\, \forall i\in {\Bbb N}$$ $N,N_i\in {\Bbb N},\,\,i=1,\ldots$, and $0<N_1<N_2<\cdots$.

Thus, we obtain an infinity branch $\displaystyle
B=\bigcup_{i=1}^NL_i$ whose leaves have the form: $$L_i=\{(r_i(z),z)\in {\Bbb
C}^2: \,z\in {\Bbb C},\,|z|>M\}.$$ Observe that we may apply this construction to any infinity point of the form $(a:b:0),\,b\not=0$.

These two approaches lead us to consider two types of infinity branches.

Let ${\cal C}$ be an affine plane curve over ${\Bbb C}$ defined by an irreducible polynomial $f(x,y) \in {\Bbb R}[x,y]$.

-   An [*infinity branch of ${\cal C}$ of type 1*]{} associated to the infinity point $P=(1:m:0),\,m\in {\Bbb C}$, 