I've looked for this data before and never found a definitive source, so this info is based on my shaky memory and is subject to correction.
Trek's lugged steel bikes had a model number system based on hundreds, where quality increased as the numbers increased. 400 series models were usually tre tubi entry level machines (I had one once and loved it), 500's were Ishwata tubed bikes, can't remember what 600's were, 700's were top quality all Reynolds 531 bikes, and the serious racer boy bikes were 800's, both in Reynolds and Columbus tubing. Within each series, models were differentiated by the second digit. 10's were sport touring, 20's were full out touring, and 30's were (again I'm shaky on this one) full out racers. Thus my old 720 was a full Reynolds triple crank touring rig. My buddy's 710 was an all 531 sport touring rig. My friend the shop owner's racer, with white paint and with decals on both down tube and seat tube, was an 830.
Paint on 400's was monochromatic metallic with no panels for the decals. 700s were also metallic but with contrasting head tubes and a seat tube panel encompassing the decal. There was one set of Trek decals on the seat tube of all models, except for the 800's that added a down tube decal and panel. The model number was indicated by a decal on the top tube near the head lug. Tubing stickers were applied near the top of the seat tube, and fork stickers were applied if the bike's fork were of high grade tubing.
If your bike has decals only on the seat tube, and is not white, I would guess it is a 700 series model. Based on the eyelets, Grand Sport components and 5 speed spacing, I'd vote for a 710.
This goes for early lugged steel Treks, before Trek went to seat lugs with cast in socketed fittings for the seat stays in the mid eighties. The numbering system still carried over after that change, but may have been modified.
Is her Trek a good riding bike? Most old steel Treks I've seen have been, in spite of the fact that they get little respect. It might be a surprise to our European friends, who perhaps are familiar only with Lance's Carbon fibre machines, to know that Trek started out making lovely lugged steel bikes back in the day. I'm having what I think is a 710 restored even as we speak.
I bought a steel lugged Trek frame for my wife in 1982. It is made with 531
throughout and has Camp dropouts with eyelets. The decals are only on the
seat tube and are separate block letters "Trek". Workmanship is good on the
frame. It has Camp cable quides on top of the BB and a stop for the clamp on
downtube shifters. It has braze on top tube brake cable guides. Rear spacing
is 120. Serial # H1C7851. Cost for the frame in 1982 was $450 new.
My question is......does anyone know the model # of this frame? I seem to remember 800 or hundred something.
By the way it is built with Gran Sport and is straight and true with original paint.