Tornado FAQs:
http://www.spc.noaa.gov/faq/tornado/

Hurricane FAQs
http://www.faqs.org/faqs/meteorology/storms-faq/part1/

Tornado FAQs
:http://www.spc.noaa.gov/faq/tornado/

From the second link

Do hurricanes and tropical storms produce tornadoes?

Often, but not always.

There are great differences from storm to storm, not necessarily
related to tropical cyclone size or intensity.

Some landfalling hurricanes in the U.S. fail to produce any known
tornadoes, while others cause major outbreaks. The same hurricane also
may have none for awhile, then erupt with tornadoes...or vice versa!

Andrew (1992), for example, spawned several tornadoes across the Deep
South after crossing the Gulf, but produced none during its rampage
across South Florida.

Katrina (2005) spawned numerous tornadoes after its devastating LA/MS
landfall, but only one in Florida (in the Keys).

Though fewer tornadoes tend to occur with tropical depressions and
tropical storms than hurricanes, there are notable exceptions like TS
Beryl of 1994 in the Carolinas.

Some tropical cyclones even produce two distinct sets of tornadoes --
one around the time of landfall over Florida or the Gulf Coast, the
other when well inland or exiting the Atlantic coast.

What's the nature of tornadoes in hurricanes and tropical storms?

Hurricane-spawned tornadoes tend to occur in small, low-topped
supercells within the outer bands, NNW through ESE of the center --
mainly the northeast quadrant.

There, the orientation and speed of the winds create vertical shear
profiles somewhat resembling those around classic Great Plains
supercells -- the shear being in a shallower layer but often stronger.

Occasionally a tornado will happen in the inner bands as well, but the
large majority still form outside the hurricane force wind zone.

Because tornado-producing circulations in hurricane supercells tend to
be smaller and shorter-lived than their Midwest counterparts, they are
harder to detect on Doppler radar, and more difficult to warn for.

Hurricane-spawned tornadoes can still be quite deadly and destructive,
as shown by the F3 tornado from Hurricane Andrew at La Place LA (1992,
2 killed) and an F4 tornado at Galveston TX from Hurricane Carla
(1961, 8 killed).

Do tropical cyclones produce waterspouts?

Yes.
Waterspouts have been observed in tropical systems. We don't know how
many of them happen in tropical cyclones but a majority probably are
from super-cells.

The similarity in Doppler radar velocity signatures over water to
tornado-producing cells in land-falling hurricanes suggests that it
may be common.

Does tropical cyclone strength or size matter for tornadoes?

Often, but not always.

Relatively weak hurricanes like Danny (1985) have spawned significant
super-cell tornadoes well inland, as have larger, more intense storms
like Beulah (1967) and Ivan (2004).

In general, the bigger and stronger the wind fields with a tropical
cyclone, the bigger the area of favourable wind shear for super-cells
and tornadoes.

Supercell tornadoes depend on instability, lift and moisture.

Surface moisture isn't lacking in a tropical cyclone but sometimes
instability and lift are too weak. This is why tropical systems tend
to produce more tornadoes in the daytime and near any fronts that may
get involved in the cyclone circulation.

It is also why SPC won't always have tornado watches out for every
instance of a tropical cyclone affecting land.

Do hurricanes and tropical storms produce tornadoes?

Andrew (1992) spawned several tornadoes across the Deep South
but produced none during its rampage across South Florida.

Katrina (2005) spawned numerous tornadoes after its devastating LA/MS
landfall, but only one in Florida (in the Keys).

It will be interesting to check how many Aleutian earthquakes occurred
when tropical storms that produced tornadoes are compared to those
that did not.

Of course there is no reason to suppose there will be more or less for
one or the other. Though I dare say that when you have tropical storms
arriving (or later, leaving) the incidence of quakes is inversely
proportional to the incidence tornadoes.