Gdtyjr

Here’s a solution that uses the excellent forest package.

documentclass[tikz,border=5pt]standalone
usepackageforest

% Node shape adapted from http://www.texample.net/tikz/examples/data-flow-diagram/
makeatletter pgfdeclareshapemyunderline
 inheritsavedanchors[from=rectangle]
 inheritanchorborder[from=rectangle]
 foreach from in
 center,base,north,north east,east,south east,south,south west,west,north west
 inheritanchor[from=rectangle]from
 
 backgroundpath
 southwest pgf@xa=pgf@x pgf@ya=pgf@y
 northeast pgf@xb=pgf@x pgf@yb=pgf@y
 % This can be improved by removing magic numbers
 pgfpathmovetopgfpointpgf@xapgf@ya+1.75em
 pgfpathlinetopgfpointpgf@xbpgf@ya+1.75em
 
 makeatother

begindocument
beginforest
for tree=
 fit=band, % Isolates space above this node from siblings’ descendants
 no edge,
 % Uncomment the line below for the dotted edges
 % edge=dotted, semithick, gray!50, shorten <=8pt, parent anchor=north,
 % This can be improved by reducing space between levels where edges are drawn
 inner sep=0pt, outer sep=0pt,
 l sep=0pt, s sep=6pt, text depth=0.5em, grow'=north,
 where level=0 % No style for dummy root node
 where n children=0
 font=bfseries,tier=word % Leaves in bold on the same tier
 font=small,tikz=node[draw, thick, myunderline, fit to=tree] ; % Non-leaves
 

% This can be improved by removing the need for a parent and sibling of the actual root
[,phantom[,phantom][OP
 [S/SN
 [Det [La] ]
 [N/Sust [novela] ]
 [CN/SAdj/Prop Sub Adj
 [PV/SV,
 [textitnexo [que] ]
 [CI/SN [me] ]
 [N/V [ha regalado] ]
 ]
 [S/SN
 [Det [mi] ]
 [N [hermana] ]
 ]
 ]
 ]
 [PN/SV
 [N/V [est'a] ]
 [Attrib/SAdj
 [N [ambientada] ]
 ]
 [CCL/SPrep
 [E [en] ]
 [T/SN
 [N [Australiarlap.] ]
 ]
 ]
 ]
]]
endforest
enddocument

And here’s another version with faint dotted edges:

You can render the same structure in a more conventional appearance just by changing options:

for tree=
 edge=dotted, semithick, gray!80, shorten <=1pt,shorten >=3pt,
 parent anchor=south, child anchor=north,
 inner sep=0pt, outer ysep=2pt,
 text depth=0.5em,
 where n children=0font=bfseries,tier=wordfont=small

So you can see why one might prefer using forest instead of bussproofs or semantics. Also, the forest tree syntax is much simpler, and is not “backwards” as seen in cfr’s answer.

Take a look at the forest manual for more style options.

2019 edit:
fit to tree option syntax has been modified to fit to=tree

For example:

documentclassstandalone
usepackage[inference]semantic
usepackage[utf8]inputenc
usepackage[T1]fontenc

begindocument

setpremisesend0pt
setpremisesspace1pt
setnamespace0pt
inference%
 inference%
 inferencemboxLaDet
 &
 inferencemboxnovelaN/Sust
 &
 inference%
 inference%
 inferencemboxquenexo
 &
 inferencemboxmeCI/SN
 &
 inferencemboxha regaladoN/V
 
 PV/SV
 &
 inference%
 inferencemboxmiDet
 &
 inferencemboxhermanaN
 
 S/SN
 
 CN/SAdj/Prop. Sub. Adj
 
 S/SN
 &
 inference%
 inferencemboxestáN/V
 &
 inference%
 inferencemboxambientadaN
 
 Attrib/SAdj
 &
 inference%
 inferencemboxenE
 &
 inference%
 inferencemboxAustraliaN
 
 T/SN
 
 CCL/SPrep
 
 PN/SV

OP

enddocument

Here's a different way... based on my answer at Highlighting text through stacked colored underlines

documentclassarticle
usepackagestackengine
newlengthtmpln
newlengthlunderset
newlengthrulethick
lunderset=1baselineskiprelax
rulethick=.8ptrelax
defstackalignmentl
newcommandnunderline[3][1]setbox0=hbox#2tmpln=wd0%
 stackunder[#1lunderset-rulethick]strut#2%
 smashraisebox-.6baselineskipmakebox[0pt][l]scriptsize #3%
 ruletmplnrulethick%
letNunnunderline
letHShspace
begindocument
 NunNunNun[3]LaDet%
 NunNun[4] %
 NunNunNun[3]novelaN/Sust%
 NunNun[4] %
NunNunNunNunNun queitshape nexo%
 NunNunNunNun[2] %
NunNunNunNunNun meCI/SN%
 NunNunNunNun[2] PV/SVCN/SAdj/Prop. Sub. AdjS/SN%
NunNunNunNunNun ha%
NunNunNunNunNun %
NunNunNunNunNun regaladoN/V%
 NunNunNun[3] %
NunNunNunNunNun miDet%
 NunNunNunNun[2] %
NunNunNunNunNun hermanaHS5exNS/SNOP%
 Nun[5] %
 NunNunNun[3]est'aN/V%
 NunNun[4] %
 NunNunNunNun[2]ambientadaHS7exNAttrib/SAdjHS7exPN/SV%
 NunNun[4] %
 NunNunNunNun[2]enEHS4exCCL/SPrep%
 NunNunNun[3] %
NunNunNunNunNun AustraliaHS6exNHS3exT/SN%
enddocument

For completeness, and in partial response to a query in the comments on my other answer, here is the tree set with prftree. As can be seen, this no more lends itself to typesetting this kind of tree than semantic. If anybody wants to typeset natural deduction proofs of this kind for logic, though, it looks like a really nice package with a lot of convenience commands, flexible customisation and a nice syntax. The tree here is very verbose because none of the convenience commands are appropriate: this is definitely a package intended for symbolic logic. The result is very, very slightly closer to the target tree in some ways than with semantic but the improvement is marginal, at best, and the result is even further from the target in other aspects.

My conclusion is that none of bussproofs, semantic and prftree are well-suited to typesetting this kind of tree. The forest and stackengine solutions are definitely superior relative to the question desiderata.

However, for logic, these package are superior to the other solutions offered here. They are designed for that purpose and will produce trees more easily and with less hassle. prftree looks especially good, even though this is the first time I've used it. The alignments and spacings are designed to work correctly out-of-the-box for logic, and prftree allows you to specify the tree using standard rules of inference, which makes it more readable, less verbose and more intuitive. My conclusion above is not, therefore, intended as a criticism of these packages. They just aren't designed for this sort of tree.

documentclass[landscape]article
usepackage[utf8]inputenc
usepackage[T1]fontenc
usepackageprftree,geometry,mathtools
begindocument

[
prftree%
 prftree%
 prftreetextLaDet
 %
 prftreetextnovelaN/Sust
 %
 prftree%
 prftree%
 prftreetextquenexo
 %
 prftreetextmeCI/SN
 %
 prftreetextha regaladoN/V
 
 PV/SV
 %
 prftree%
 prftreetextmiDet
 %
 prftreetexthermanaN
 S/SN
 
 CN/SAdj/Prop. Sub. Adj
 
 S/SN
 
 prftree%
 prftreetextestáN/V
 %
 prftree%
 prftreetextambientadaN
 
 Attrib/SAdj
 %
 prftree%
 prftreetextenE
 %
 prftree%
 prftreetextAustraliaN
 
 T/SN
 
 CCL/SPrep
 
 PN/SV

OP
]

enddocument