The vertical signal path of the 581 is the same as that of the 585. It uses the T581 CRT. Type 581 scopes with serial numbers up to 510 use an all-tube trigger cicrcuit. Starting at serial number 511, the 581 uses tunnel diode triggering.
A Type 581A was introduced later, and it is specified to have a bandwidth of 85MHz instead of 100MHz. The somewhat relaxed bandwidth specification improved CRT production yield which had been a problem when the scope was specified to 100MHz.