If you can use xslt2 then it's just

<xsl:stylesheet version="2.0" xmlns:xsl="http://www.w3.org/1999/XSL/Transform">
  
<xsl:template match="/*">
  <xsl:for-each-group select=".//*" group-by="string-join(ancestor-or-self::*/name(),'/')">
    <xsl:value-of select="'&#10;(',count(current-group()),') ',current-grouping-key()" separator=""/>
  </xsl:for-each-group>
</xsl:template>
</xsl:stylesheet>

which (if you put <x>...</x> around your posted input to make it well
formed) produces:

$ saxon8 depth.xml depth.xsl
<?xml version="1.0" encoding="UTF-8"?>
(2) x/a1
(2) x/a1/b1
(2) x/a1/b1/c1
(1) x/a1/b1/c2
(1) x/a1/b1/c3
(2) x/a1/b2
(1) x/a1/b2/c2
(1) x/a1/b2/c3
(2) x/a1/b3
(1) x/a1/b3/c1
(1) x/a1/b3/c4


David