Count number of directories

LC_ALL=C find .//. -name 'lib*' -type d | grep -c //

You can't use find . | wc -l as that wouldn't work properly if there are file paths with newline characters.

Without LC_ALL=C that could fail to count dir names that start with lib but where the rest of the name contains bytes that don't form valid characters.

With zsh:

(){echo $#} **/lib*(DN/)

Assuming all lib-directories have sane names with no newlines or other strange/exotic characters:

find / -type d -name "lib*" -print | wc -l

This assumes that you, by the "whole directory subtree", mean "anywhere". Change / into . to only count in the current directory or below.

The find command would find all directories (-type d) whose name start with lib (-name "lib*") and print these (-print).

The wc -l would count the number of lines in the output from find.

Here's a bash-only solution, using globstar shell option:

$ shopt -s globstar; c=0; for f in ** ; do echo "${f##*/}"; [[ "${f##*/}" =~ ^lib* ]] && [[ -d "$f" ]]  && ((c++)); done ; echo $c

Or in script format for readability:

#!/usr/bin/env bash
shopt -s globstar 
c=0
for f in **
do 
    echo "${f##*/}"
    if [[ "${f##*/}" =~ ^lib* ]] && [[ -d "$f" ]]
    then 
        ((c++))
    done
echo $c

globstar allows us to enable recursive globbing and via [[ with pattern matching and -d flag to check if that's a directory, we increment the counter variable $c, which we print at the end.