  The determination of the braid index of an oriented link is generally a hard
problem. In the case of alternating links, some significant progresses have
been made in recent years which made explicit and precise braid index
computations possible for links from various families of alternating links,
including the family of all alternating Montesinos links. However, much less is
known for non-alternating links. For example, even for the non-alternating
pretzel links, which are special (and simpler) Montesinos links, the braid
index is only known for a very limited few special cases. In this paper and its
sequel, we study the braid indices for all non-alternating pretzel links by a
systematic approach. We classify the pretzel links into three different types
according to the Seifert circle decompositions of their standard link diagrams.
More specifically, if $D$ is a standard diagram of an oriented pretzel link
$\mathcal{L}$, $S(D)$ is the Seifert circle decomposition of $D$, and $C_1$,
$C_2$ are the Seifert circles in $S(D)$ containing the top and bottom long
strands of $D$ respectively, then $\mathcal{L}$ is classified as a Type 1 (Type
2) pretzel link if $C_1\not=C_2$ and $C_1$, $C_2$ have different (identical)
orientations. In the case that $C_1=C_2$, then $\mathcal{L}$ is classified as a
Type 3 pretzel link. In this paper, we present the results of our study on Type
1 and Type 2 pretzel links. Our results allow us to determine the precise braid
index for any non-alternating Type 1 or Type 2 pretzel link. Since the braid
indices are already known for all alternating pretzel links from our previous
work, it means that we have now completely determined the braid indices for all
Type 1 and Type 2 pretzel links.
